Approximate Bayesian Inference Research Articles

Bayesian Automatic Relevance Determination for Utility Function Specification in Discrete Choice Models

Specifying utility functions is a key step towards applying the discrete choice framework for understanding the behaviour processes that govern user choices. However, identifying the utility function specifications that best model and explain the observed choices can be a very challenging and time-consuming task. This paper seeks to help modellers by leveraging the Bayesian framework and the concept of automatic relevance determination (ARD), in order to automatically determine an optimal utility function specification from an exponentially large set of possible specifications in a purely data-driven manner. Based on recent advances in approximate Bayesian inference, a doubly stochastic variational inference is developed, which allows the proposed MNL-ARD model to scale to very large and high-dimensional datasets. Using semi-artificial choice data, the proposed approach is shown to be able to accurately recover the true utility function specifications that govern the observed choices. Moreover, when applied to real choice data, MNL-ARD is able discover high quality specifications that can outperform previous ones from the literature according to multiple criteria, thereby demonstrating its practical applicability.

Deep Active Inference and Scene Construction.

Adaptive agents must act in intrinsically uncertain environments with complex latent structure. Here, we elaborate a model of visual foraging—in a hierarchical context—wherein agents infer a higher-order visual pattern (a “scene”) by sequentially sampling ambiguous cues. Inspired by previous models of scene construction—that cast perception and action as consequences of approximate Bayesian inference—we use active inference to simulate decisions of agents categorizing a scene in a hierarchically-structured setting. Under active inference, agents develop probabilistic beliefs about their environment, while actively sampling it to maximize the evidence for their internal generative model. This approximate evidence maximization (i.e., self-evidencing) comprises drives to both maximize rewards and resolve uncertainty about hidden states. This is realized via minimization of a free energy functional of posterior beliefs about both the world as well as the actions used to sample or perturb it, corresponding to perception and action, respectively. We show that active inference, in the context of hierarchical scene construction, gives rise to many empirical evidence accumulation phenomena, such as noise-sensitive reaction times and epistemic saccades. We explain these behaviors in terms of the principled drives that constitute the expected free energy, the key quantity for evaluating policies under active inference. In addition, we report novel behaviors exhibited by these active inference agents that furnish new predictions for research on evidence accumulation and perceptual decision-making. We discuss the implications of this hierarchical active inference scheme for tasks that require planned sequences of information-gathering actions to infer compositional latent structure (such as visual scene construction and sentence comprehension). This work sets the stage for future experiments to investigate active inference in relation to other formulations of evidence accumulation (e.g., drift-diffusion models) in tasks that require planning in uncertain environments with higher-order structure.

JTF Analysis of Micromotion Targets Based on Single-Window Variational Inference

This article addresses the problem of joint time–frequency (JTF) analysis of micromotion targets in complex environments in an approximate Bayesian inference framework. First, the sparse observation model is constructed, which is then decomposed into a series of single-window-JTF (SW-JTF) analysis problems to tackle the high dimension of the over-complete dictionary. On this basis, the probabilistic graphical model is constructed by imposing the Gamma-complex Gaussian prior to the JTF distribution. Finally, the model parameters are solved effectively by single-window variational inference (SWVI). Compared with the available methods, the proposed method could obtain better-focused JTF signature for narrowband data and higher quality range-instantaneous Doppler (RID) image for wideband data, especially in low signal-to-noise ratio (SNR) and data corruption scenarios.

Uncertainty modelling in deep learning for safer neuroimage enhancement: Demonstration in diffusion MRI

Deep learning (DL) has shown great potential in medical image enhancement problems, such as super-resolution or image synthesis. However, to date, most existing approaches are based on deterministic models, neglecting the presence of different sources of uncertainty in such problems. Here we introduce methods to characterise different components of uncertainty, and demonstrate the ideas using diffusion MRI super-resolution. Specifically, we propose to account for intrinsic uncertainty through a heteroscedastic noise model and for parameter uncertainty through approximate Bayesian inference, and integrate the two to quantify predictive uncertainty over the output image. Moreover, we introduce a method to propagate the predictive uncertainty on a multi-channelled image to derived scalar parameters, and separately quantify the effects of intrinsic and parameter uncertainty therein. The methods are evaluated for super-resolution of two different signal representations of diffusion MR images—Diffusion Tensor images and Mean Apparent Propagator MRI—and their derived quantities such as mean diffusivity and fractional anisotropy, on multiple datasets of both healthy and pathological human brains. Results highlight three key potential benefits of modelling uncertainty for improving the safety of DL-based image enhancement systems. Firstly, modelling uncertainty improves the predictive performance even when test data departs from training data (“out-of-distribution” datasets). Secondly, the predictive uncertainty highly correlates with reconstruction errors, and is therefore capable of detecting predictive “failures”. Results on both healthy subjects and patients with brain glioma or multiple sclerosis demonstrate that such an uncertainty measure enables subject-specific and voxel-wise risk assessment of the super-resolved images that can be accounted for in subsequent analysis. Thirdly, we show that the method for decomposing predictive uncertainty into its independent sources provides high-level “explanations” for the model performance by separately quantifying how much uncertainty arises from the inherent difficulty of the task or the limited training examples. The introduced concepts of uncertainty modelling extend naturally to many other imaging modalities and data enhancement applications.

Approximate Bayesian inference for case-crossover models.

A case-crossover analysis is used as a simple but powerful tool for estimating the effect of short-term environmental factors such as extreme temperatures or poor air quality on mortality. The environment on the day of each death is compared to the one or more "control days" in previous weeks, and higher levels of exposure on death days than control days provide evidence of an effect. Current state-of-the-art methodology and software (integrated nested Laplace approximation [INLA]) cannot be used to fit the most flexible case-crossover models to large datasets, because the likelihood for case-crossover models cannot be expressed in a manner compatible with this methodology. In this paper, we develop a flexible and scalable modeling framework for case-crossover models with linear and semiparametric effects which retains the flexibility and computational advantages of INLA. We apply our method to quantify nonlinear associations between mortality and extreme temperatures in India. An R package implementing our methods is publiclyavailable.

High-Resolution Radar Imaging in Low SNR Environments Based on Expectation Propagation

We address the problem of high-resolution radar imaging in low signal-to-noise ratio (SNR) environments in an approximate Bayesian inference framework. First, the probabilistic graphical model is constructed by imposing the sparsity-promoting spike-and-slab prior to the distribution of scattering centers. Then, the model parameters and phase errors are estimated iteratively by expectation propagation (EP) and maximum likelihood (ML) estimation. Compared with the available imaging methods based on the numerical optimization and Bayesian inference, the proposed method has exhibited more flexibility in data representation and better performance in parameter estimation, particularly in sparse-aperture and low SNR scenarios.

Approximate Bayesian inference of seismic velocity and pore-pressure uncertainty with basin modeling, rock physics, and imaging constraints

We have introduced a methodology for quantifying seismic velocity and pore-pressure uncertainty that incorporates information regarding the geologic history of a basin, rock physics, well log, drilling, and seismic data. In particular, our approach relies on linking velocity models to the basin modeling outputs of porosity, mineral volume fractions, and pore pressure through rock-physics models. We account for geologic uncertainty by defining prior probability distributions on lithology-specific porosity compaction model parameters, permeability-porosity model parameters, and heat-flow boundary condition. Monte Carlo basin simulations are performed by sampling the prior uncertainty space. We perform probabilistic calibration of the basin model outputs by defining data likelihood distributions to represent well data uncertainty. Rock physics modeling transforms the basin modeling outputs to give us multiple velocity realizations used to perform multiple depth migrations. We have developed an approximate Bayesian inference framework that uses migration velocity analysis in conjunction with well data for updating velocity and basin modeling uncertainty. We apply our methodology in 2D to a real field case from the Gulf of Mexico; our methodology allows for building a geologic and physical model space for velocity and pore-pressure prediction with reduced uncertainty.

Max-and-Smooth: A Two-Step Approach for Approximate Bayesian Inference in Latent Gaussian Models

With modern high-dimensional data, complex statistical models are necessary, requiring computationally feasible inference schemes. We introduce Max-and-Smooth, an approximate Bayesian inference scheme for a flexible class of latent Gaussian models (LGMs) where one or more of the likelihood parameters are modeled by latent additive Gaussian processes. Our proposed inference scheme is a two-step approach. In the first step (Max), the likelihood function is approximated by a Gaussian density with mean and covariance equal to either (a) the maximum likelihood estimate and the inverse observed information, respectively, or (b) the mean and covariance of the normalized likelihood function. In the second step (Smooth), the latent parameters and hyperparameters are inferred and smoothed with the approximated likelihood function. The proposed method ensures that the uncertainty from the first step is correctly propagated to the second step. Because the prior density for the latent parameters is assumed to be Gaussian and the approximated likelihood function is Gaussian, the approximate posterior density of the latent parameters (conditional on the hyperparameters) is also Gaussian, thus facilitating efficient posterior inference in high dimensions. Furthermore, the approximate marginal posterior distribution of the hyperparameters is tractable, and as a result, the hyperparameters can be sampled independently of the latent parameters. We show that the computational cost of Max-and-Smooth is close to being insensitive to the number of independent data replicates, and that it scales well with increased dimension of the latent parameter vector provided that its Gaussian prior density is specified with a sparse precision matrix. In the case of a large number of independent data replicates, sparse precision matrices, and high-dimensional latent vectors, the speedup is substantial in comparison to an MCMC scheme that infers the posterior density from the exact likelihood function. The accuracy of the Gaussian approximation to the likelihood function increases with the number of data replicates per latent model parameter. The proposed inference scheme is demonstrated on one spatially referenced real dataset and on simulated data mimicking spatial, temporal, and spatio-temporal inference problems. Our results show that Max-and-Smooth is accurate and fast.

Stochastic dynamics of Francisella tularensis infection and replication.

We study the pathogenesis of Francisella tularensis infection with an experimental mouse model, agent-based computation and mathematical analysis. Following inhalational exposure to Francisella tularensis SCHU S4, a small initial number of bacteria enter lung host cells and proliferate inside them, eventually destroying the host cell and releasing numerous copies that infect other cells. Our analysis of disease progression is based on a stochastic model of a population of infectious agents inside one host cell, extending the birth-and-death process by the occurrence of catastrophes: cell rupture events that affect all bacteria in a cell simultaneously. Closed expressions are obtained for the survival function of an infected cell, the number of bacteria released as a function of time after infection, and the total bacterial load. We compare our mathematical analysis with the results of agent-based computation and, making use of approximate Bayesian statistical inference, with experimental measurements carried out after murine aerosol infection with the virulent SCHU S4 strain of the bacterium Francisella tularensis, that infects alveolar macrophages. The posterior distribution of the rate of replication of intracellular bacteria is consistent with the estimate that the time between rounds of bacterial division is less than 6 hours in vivo.

Learning Demand Curves in B2B Pricing: A New Framework and Case Study

In business‐to‐business (B2B) pricing, a seller seeks to maximize revenue obtained from high‐volume transactions involving a wide variety of buyers, products, and other characteristics. Buyer response is highly uncertain, and the seller only observes whether buyers accept or reject the offered prices. These deals are also subject to high opportunity cost, since revenue is zero if the price is rejected. The seller must adapt to this uncertain environment and learn quickly from new deals as they take place. We propose a new framework for statistical and optimal learning in this problem, based on approximate Bayesian inference, which has the ability to measure and update the seller’s uncertainty about the demand curve based on new deals. In a case study, based on historical data, we show that our approach offers significant practical benefits.

Anomaly Detection System for Water Networks in Northern Ethiopia Using Bayesian Inference

For billions of people living in remote and rural communities in the developing countries, small water systems are the only source of clean drinking water. Due to the rural nature of such water systems, site visits may occur infrequently. This means broken water systems can remain in a malfunctioning state for months, forcing communities to return to drinking unsafe water. In this work, we present a novel two-level anomaly detection system aimed to detect malfunctioning remote sensored water hand-pumps, allowing for a proactive approach to pump maintenance. To detect anomalies, we need a model of normal water usage behavior first. We train a multilevel probabilistic model of normal usage using approximate variational Bayesian inference to obtain a conditional probability distribution over the hourly water usage data. We then use this conditional distribution to construct a level-1 scoring function for each hourly water observation and a level-2 scoring function for each pump. Probabilistic models and Bayesian inference collectively were chosen for their ability to capture the high temporal variability in the water usage data at the individual pump level as well as their ability to estimate interpretable model parameters. Experimental results in this work have demonstrated that the pump scoring function is able to detect malfunctioning sensors as well as a change in water usage behavior allowing for a more responsive and proactive pump system maintenance.

Evaluating Scalable Uncertainty Estimation Methods for Deep Learning-Based Molecular Property Prediction.

Advances in deep neural network (DNN)-based molecular property prediction have recently led to the development of models of remarkable accuracy and generalization ability, with graph convolutional neural networks (GCNNs) reporting state-of-the-art performance for this task. However, some challenges remain, and one of the most important that needs to be fully addressed concerns uncertainty quantification. DNN performance is affected by the volume and the quality of the training samples. Therefore, establishing when and to what extent a prediction can be considered reliable is just as important as outputting accurate predictions, especially when out-of-domain molecules are targeted. Recently, several methods to account for uncertainty in DNNs have been proposed, most of which are based on approximate Bayesian inference. Among these, only a few scale to the large data sets required in applications. Evaluating and comparing these methods has recently attracted great interest, but results are generally fragmented and absent for molecular property prediction. In this paper, we quantitatively compare scalable techniques for uncertainty estimation in GCNNs. We introduce a set of quantitative criteria to capture different uncertainty aspects and then use these criteria to compare MC-dropout, Deep Ensembles, and bootstrapping, both theoretically in a unified framework that separates aleatoric/epistemic uncertainty and experimentally on public data sets. Our experiments quantify the performance of the different uncertainty estimation methods and their impact on uncertainty-related error reduction. Our findings indicate that Deep Ensembles and bootstrapping consistently outperform MC-dropout, with different context-specific pros and cons. Our analysis leads to a better understanding of the role of aleatoric/epistemic uncertainty, also in relation to the target data set features, and highlights the challenge posed by out-of-domain uncertainty.

Correcting Predictions for Approximate Bayesian Inference

Bayesian models quantify uncertainty and facilitate optimal decision-making in downstream applications. For most models, however, practitioners are forced to use approximate inference techniques that lead to sub-optimal decisions due to incorrect posterior predictive distributions. We present a novel approach that corrects for inaccuracies in posterior inference by altering the decision-making process. We train a separate model to make optimal decisions under the approximate posterior, combining interpretable Bayesian modeling with optimization of direct predictive accuracy in a principled fashion. The solution is generally applicable as a plug-in module for predictive decision-making for arbitrary probabilistic programs, irrespective of the posterior inference strategy. We demonstrate the approach empirically in several problems, confirming its potential.

Few-Shot Bayesian Imitation Learning with Logical Program Policies

Humans can learn many novel tasks from a very small number (1–5) of demonstrations, in stark contrast to the data requirements of nearly tabula rasa deep learning methods. We propose an expressive class of policies, a strong but general prior, and a learning algorithm that, together, can learn interesting policies from very few examples. We represent policies as logical combinations of programs drawn from a domain-specific language (DSL), define a prior over policies with a probabilistic grammar, and derive an approximate Bayesian inference algorithm to learn policies from demonstrations. In experiments, we study six strategy games played on a 2D grid with one shared DSL. After a few demonstrations of each game, the inferred policies generalize to new game instances that differ substantially from the demonstrations. Our policy learning is 20–1,000x more data efficient than convolutional and fully convolutional policy learning and many orders of magnitude more computationally efficient than vanilla program induction. We argue that the proposed method is an apt choice for tasks that have scarce training data and feature significant, structured variation between task instances.

On the XUV Luminosity Evolution of TRAPPIST-1

Abstract We model the long-term X-ray and ultraviolet (XUV) luminosity of TRAPPIST-1 to constrain the evolving high-energy radiation environment experienced by its planetary system. Using a Markov Chain Monte Carlo (MCMC) method, we derive probabilistic constraints for TRAPPIST-1's stellar and XUV evolution that account for observational uncertainties, degeneracies between model parameters, and empirical data of low-mass stars. We constrain TRAPPIST-1's mass to m ⋆ = 0.089 ± 0.001 M ⊙ and find that its early XUV luminosity likely saturated at . From the posterior distribution, we infer that there is a ∼40% chance that TRAPPIST-1 is still in the saturated phase today, suggesting that TRAPPIST-1 has maintained high activity and L XUV/L bol ≈ 10−3 for several gigayears. TRAPPIST-1's planetary system therefore likely experienced a persistent and extreme XUV flux environment, potentially driving significant atmospheric erosion and volatile loss. The inner planets likely received XUV fluxes ∼103–104 times that of the modern Earth during TRAPPIST-1's ∼1 Gyr long pre-main-sequence phase. Deriving these constraints via MCMC is computationally nontrivial, so scaling our methods to constrain the XUV evolution of a larger number of M dwarfs that harbor terrestrial exoplanets would incur significant computational expenses. We demonstrate that approxposterior, an open source Python machine learning package for approximate Bayesian inference using Gaussian processes, accurately and efficiently replicates our analysis using 980 times less computational time and 1330 times fewer simulations than MCMC sampling using emcee. We find that approxposterior derives constraints with mean errors on the best-fit values and 1σ uncertainties of 0.61% and 5.5%, respectively, relative to emcee.

Pakman: a modular, efficient and portable tool for approximate Bayesian inference

Pakman: a modular, efficient and portable tool for approximate Bayesian inference

Using approximate Bayesian inference for a "steps and turns" continuous-time random walk observed at regular time intervals.

The study of animal movement is challenging because movement is a process modulated by many factors acting at different spatial and temporal scales. In order to describe and analyse animal movement, several models have been proposed which differ primarily in the temporal conceptualization, namely continuous and discrete time formulations. Naturally, animal movement occurs in continuous time but we tend to observe it at fixed time intervals. To account for the temporal mismatch between observations and movement decisions, we used a state-space model where movement decisions (steps and turns) are made in continuous time. That is, at any time there is a non-zero probability of making a change in movement direction. The movement process is then observed at regular time intervals. As the likelihood function of this state-space model turned out to be intractable yet simulating data is straightforward, we conduct inference using different variations of Approximate Bayesian Computation (ABC). We explore the applicability of this approach as a function of the discrepancy between the temporal scale of the observations and that of the movement process in a simulation study. Simulation results suggest that the model parameters can be recovered if the observation time scale is moderately close to the average time between changes in movement direction. Good estimates were obtained when the scale of observation was up to five times that of the scale of changes in direction. We demonstrate the application of this model to a trajectory of a sheep that was reconstructed in high resolution using information from magnetometer and GPS devices. The state-space model used here allowed us to connect the scales of the observations and movement decisions in an intuitive and easy to interpret way. Our findings underscore the idea that the time scale at which animal movement decisions are made needs to be considered when designing data collection protocols. In principle, ABC methods allow to make inferences about movement processes defined in continuous time but in terms of easily interpreted steps and turns.

Technical Note—Consistency Analysis of Sequential Learning Under Approximate Bayesian Inference

“Bayesian learning works even with censored information” We often learn about a problem from information that is incomplete or censored: for example, a medical treatment may cause side effects with no indication of what the right dose should have been. Bayesian belief models are useful in such settings, but cannot be constructed using traditional methods; as a result, practitioners have developed ways of constructing them approximately. These approximations have been very successful in many application domains, yet until now have lacked theoretical support. The paper “Consistency analysis of sequential learning under approximate Bayesian inference,” by Chen and Ryzhov, links approximate Bayesian learning to stochastic approximation theory. Using this link, the authors prove – for the first time – the consistency of a suite of approximate Bayesian methods culled from the literature. One highlight is an entirely new consistency proof for Bayesian logistic regression, a well-established approximation technique that essentially treats logistic regression as if it were ordinary least squares.

Coalescent Simulation with msprime.

Coalescent simulation is a fundamental tool in modern population genetics. The msprime library provides unprecedented scalability in terms of both the simulations that can be performed and the efficiency with which the results can be processed. We show how coalescent models for population structure and demography can be constructed using a simple Python API, as well as how we can process the results of such simulations to efficiently calculate statistics of interest. We illustrate msprime's flexibility by implementing a simple (but functional) approximate Bayesian computation inference method in just a few tens of lines of code.

Fast approximate inference for variable selection in Dirichlet process mixtures, with an application to pan-cancer proteomics.

The Dirichlet Process (DP) mixture model has become a popular choice for model-based clustering, largely because it allows the number of clusters to be inferred. The sequential updating and greedy search (SUGS) algorithm (Wang & Dunson, 2011) was proposed as a fast method for performing approximate Bayesian inference in DP mixture models, by posing clustering as a Bayesian model selection (BMS) problem and avoiding the use of computationally costly Markov chain Monte Carlo methods. Here we consider how this approach may be extended to permit variable selection for clustering, and also demonstrate the benefits of Bayesian model averaging (BMA) in place of BMS. Through an array of simulation examples and well-studied examples from cancer transcriptomics, we show that our method performs competitively with the current state-of-the-art, while also offering computational benefits. We apply our approach to reverse-phase protein array (RPPA) data from The Cancer Genome Atlas (TCGA) in order to perform a pan-cancer proteomic characterisation of 5157 tumour samples. We have implemented our approach, together with the original SUGS algorithm, in an open-source R package named sugsvarsel, which accelerates analysis by performing intensive computations in C++ and provides automated parallel processing. The R package is freely available from: https://github.com/ococrook/sugsvarsel.